I will name the things which Roger wants in their order of importance. Firstly, Roger wants a family.
"Ain't you got nobody home to tell you to wash your face?"
"No, m'," said the boy.
It is obvious from what he says that he does not have anyone to go home to. He wants a proper home, not just a place to stay (which he already seems to have). A place in which he can feel comfortable, taken care of, and loved.
Secondly, Roger wants to be trusted. He could have run away, since the door was open, but does not.
He did not trust the woman not to trust him. And he didn't want to be mistrusted now.
Roger has probably never earned the trust of anyone, let alone a stranger, and he wanted to be trusted at this point. He wanted to prove himself better than others' suspicion.
Thirdly,Roger wants money to take of his most immediate need - hunger. He has obviously had nothing to eat even though he tells Mrs Jones otherwise.
"I believe you're hungry - or been hungry - to try to snatch my pocketbook."
And fourthly, on a more materialistic level, Roger wanted a pair of blue-suede shoes.
"I wanted a pair of blue-suede shoes," said the boy.
It is ironic that this is the reason he puts forward for wanting to steal Mrs Jones' pocketbook, since he is scruffy and dirty, so much so, that she lets him wash his face. Roger obviously wants the shoes to make him (as he believes), look better and therefore also feel better about himself.
Mrs Jones, an old woman, obviously wants somebody she can take care of.
"You ought to be my son. I can teach you right from wrong."
She, just like Roger, is without family and he could fill the emptiness in her life. In this, he could provide her with the second thing she wants, companionship. Roger could be someone with whom she shares her thought, her experiences and so on.
Both these would heal the loneliness and make her life more fulfilling. It is therefore sad that Roger 'never saw her again.'